where the term ℤ⊕ℤ\mathbb{Z} \oplus \mathbb{Z} is in degree 0: this is the free abelian group on the set {0,1}\{0,1\} of 0-simplices in Δ[1]\Delta[1]. The other copy of ℤ\mathbb{Z} is the free abelian group on the single non-degenerate edge in Δ[1]\Delta[1]. All other cells of Δ[1]\Delta[1] are degenerate and hence do not contribute to the normalized chain complex. The single nontrivial differential sends 1∈ℤ1 \in \mathbb{Z} to (1,−1)∈ℤ⊕ℤ(1,-1) \in \mathbb{Z} \oplus \mathbb{Z}, reflecting the fact that one of the vertices is the 0-boundary and the other is the 1-boundary of the single nontrivial edge.

Therefore a chain map (f,g,ψ):C•⊗I•→D•(f,g,\psi) : C_\bullet \otimes I_\bullet \to D_\bullet that restricted to the two copies of C•C_\bullet is ff and gg, respectively, is characterized by a collection of commuting diagrams

On the elements (1,0,0)(1,0,0) and (0,1,0)(0,1,0) in the top left this reduces to the chain map condition for ff and gg, respectively. On the element (0,0,1)(0,0,1) this is the equation for the chain homotopy

Remark

Beware, as discussed there, that another category that would deserve to carry this name instead is called the derived category of 𝒜\mathcal{A}. In the derived category one also quotients out chain homotopy, but one allows that first the domain of the two chain maps ff and gg is refined along a quasi-isomorphism.